Statics of beams, pulleys and the balance
Weights and equilibrium: oblique beams on cords and pulleys, a pulley train, and the law of the lever
A dense mechanics sheet crowded with diagrams of beams suspended by cords and pulleys, a balance, a five-pulley train and beams resting on inclined planes, surrounded by explanatory text. Leonardo distinguishes the two natures of weight in an oblique beam, reasons about how pulleys multiply a load, and states the law of the lever — that the lesser weight on the greater arm equals the greater weight on the lesser arm. He also notes in passing that water within water loses the weight it had in air, and that lengthening a balance arm lessens the water a connected pipe pours out. An incomplete note at the top mentions glass furnaces.
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An oblique beam has two natures of weight
A uniform beam set obliquely shows two natures of weight — one when hung by its ends from two hangers of a balance, another when laid obliquely. Leonardo argues it cannot be supported at once on the balance-hangers and on a second oblique beam, since resting on the cords gives it a greater weight than resting on the beam.
Beam on two pulleys: variable and permanent gravity
The beam a b hung from two pulleys has two kinds of gravity: one variable to infinity, the other permanent. The variable kind arises when the weight descends over a place of uniform obliquity.
A five-pulley train doubling the first weight
A system of five pulleys is annotated 2-1-2-4-8-16. The doublings of the first weight equal half the number of cords hanging from the pulleys, or the number of pulleys themselves.
The law of the lever
The lesser weight on the greater arm does as much as the greater weight on the lesser arm; weights that resist one another's descent stand as the arms of the balance, but reversed. Motion arises only when some excess is added — by shortening the lesser arm, lengthening the greater, or altering the weights.
Water loses weight in water; the pipe pours less
The same note observes that water within water loses the weight it had in air. Lengthening the greater balance arm lessens the motion of its end, and so lessens the water that the pipe pours out above.
Cord stays in line with an oblique beam
A beam disposed to slide down an oblique plane is held by a cord that always continues in a straight line with the central line of the beam. It is therefore impossible to generate an angle of any kind between cord and beam.